16.15:
Modes of Standing Waves: II
Consider air trapped inside a tube that is open at one end and closed at the other. If a wave impinges on it, it reflects, producing standing waves.
The boundary conditions are antinodes at the open end, where air molecules are free to vibrate, and nodes at the closed end, where they are not.
If both ends were closed, the first possible mode would have nodes at both ends. But here, the first possible mode can be obtained by stretching this pattern. The tube's length equals one-fourth the wavelength of the fundamental mode.
For the first overtone, the length is equal to three-fourths the wavelength. For the second overtone, it is five-fourths the wavelength, and so on.
So, the wavelengths of the standing wave modes are found to follow a mathematical pattern given by n, an odd integer.
The corresponding frequencies, or the harmonics, are obtained in terms of wave speed.
16.15:
Modes of Standing Waves: II
The starting point for expressing the modes of standing waves is understanding the boundary conditions that the waves must follow. The boundary conditions are derived from the physical understanding of how the standing waves are sustained, that is, how the vibrating particles of the medium behave at the boundaries imposed on them.
For a tube open at one end and closed at the other filled with air, the modes are such that there is always an antinode at the open end and a node at the closed end. Interestingly, the resonant frequencies depend on the speed of sound, of which depends on temperature. This can pose noticeable problems, for example, for organs in old unheated cathedrals. It is also why musicians commonly bring their wind instruments to room temperature before playing them.
The harmonics for a tube can be similarly derived. First, the boundary conditions must be noted; the air molecules are free to vibrate at both ends. As a result, both ends are antinodes. This can be used to determine the mode's wavelength which is equal to an integer multiple of half the tube's length. The harmonics follow the same mathematical pattern as that of standing waves on a string, which has nodes at both ends.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-introduction: sections 16.6 and 17.4; pages 829-830 and 872-875.